Trying to fake "God Rays"/ Stage Lights with materials
Well I think the gradient might be at least some help to you, if you hook it up to the density.  Here is an example with a linear gradient:



Here is the same image with the gradient disconnected:



It is true that if there is ambient lighting, a certain amount of volume particles may be visible:



Some of that could be mitigated by strategic repositioning of your spot lamp:



and/or dialing back the gradient, for example by altering the "location" component of the mapping node (here I set the X coordinate to -0.2):



Hopefully these give you some ideas for tweaking your setup.  (By the way, I did not end up making use of the Multiply node in the group below, but it's another parameter you could play with).